The rise of Euclidean axiomatics

Colin Maclarty

Case Western Reserve University, Cleveland

We apply ideas from Ian Mueller and David Fowler to the rise of Euclidean axiomatics. Hippocrates of Chios invokes a curvilinear case of the Pythagorean theorem about 200 years before Euclid. A Euclid-style treatment of this case would require extending proofs in Euclid's Books I, VI, and XII. The evidence suggests Hippocrates used an argument in a few lines, while Euclid I.47 takes scores of pages just for the familiar case of squares on a right triangle. Euclid's radically longer and radically more explicit Elements eclipsed all earlier geometries as utterly as Homer eclipsed all earlier epic poets, and became a paradigm framework for mathematics to this day.